Kolchin Seminar in Differential Algebra
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Academic year 2007–2008

Last updated on May 14, 2017.
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In Fall 2007, we concentrated on differential Galois (Picard-Vessiot) theory.     For a general reference click here.

Friday, August 31, 2007

Richard Churchill, The Graduate Center and Hunter College
A brief introduction to algebraic geometry, part 1

For lecture notes for all 3 lectures, click here.
For notes on constrained points, click here.

Friday, September 7, 2007

Richard Churchill, The Graduate Center and Hunter College
A brief introduction to algebraic geometry, part 2

Tuesday, September 18, 2007

Richard Churchill, The Graduate Center and Hunter College
A brief introduction to algebraic geometry, part 3

Friday, September 28, 2007

Anand Pillay, University of Leeds
Transcendence questions for logarithmic differential equations on nonconstant semiabelian varieties
(Joint with D. Bertrand)

Friday, October 5, 2007

Phyllis Joan Cassidy, Smith College and City College
Introduction to differential algebraic geometry and differential algebraic groups

Differential algebraic geometry is a new geometry, invented and developed by Ritt and Kolchin, that seeks to understand and clarify the approach to algebraic differential equations taken by Picard, Vessiot, Painlevé, and others at the turn of the 20th century. We will first spend some time in the Weil-based context of Kolchin, and then move to the contemporary Grothendieck-based context of Kovacic, Keigher, and others.

For lecture notes click here.

Friday, October 12, 2007

David Marker, University of Illinois at Chicago
Manin Kernels

Manin proved that there are interesting differential-algebraic homomorphisms from Abelian varieties to vector groups. The kernels of these maps geive rise to finite dimensional differential-algebraic subgroups that arise in number theoretic applications of Buium and Hrushovski. I will survey the construction of these Manin kernels and describe some of their model theoretic properties.

Friday, October 19, 2007

Phyllis Joan Cassidy, Smith College and City College
Introduction to differential algebraic geometry and differential algebraic groups, part 2

For lecture notes click here.

Friday, October 26, 2007

Phyllis Joan Cassidy, Smith College and City College
Introduction to differential algebraic geometry and differential algebraic groups, part 3


For lecture notes, click here.

Friday, November 2, 2007

Phyllis Joan Cassidy, Smith College and City College
Introduction to differential algebraic geometry and differential algebraic groups, part 4


For lecture notes, click here.

Friday, November 9, 2007

Phyllis Joan Cassidy, Smith College and City College
Introduction to differential algebraic geometry and differential algebraic groups, part 5


For lecture notes, click here.

Friday, November 16, 2007

Phyllis Joan Cassidy, Smith College and City College
Differentially closed fields

We will survey the concept of differentially closed fields of characteristic 0, within the framework of Kolchin's differential algebraic geometry. We will also discuss variations in the \mathbb{Z}-rank of Kolchin and Manin kernels on elliptic curves whose points lie in a differentially closed field. Kolchin introduced the kernel of the logarithmic derivative homomorphism on an elliptic curve defined over constants. Manin introduced the kernel of a homomorphism constructed from the Picard-Fuchs differential equation on an elliptic curve defined over the function field \mathbb{C}(t) that does not descend to constants. In the discussion of the Manin kernel, we will refer to results of Manin, Buium, Hrushovski, and Pillay.


For lecture notes, click here.
For an addendum on the logarithmic derivative, click here.

Friday, November 30, 2007

Jerald Kovacic, City College
Differential schemes

We start with the definition of differential scheme using the treatment of Hartshorne. Immediately we find that there are problems: the global section functor of an affine differential scheme does not recover the original ring. We give some examples to show what goes wrong. We shall also discuss what is known and not known.

For lecture notes click here.

Friday, December 7, 2007

Jerald Kovacic, City College
Differential schemes, part 2

For lecture notes click here.

Friday, January 25, 2008

Jerald Kovacic, City College
Strongly normal extensions

This lecture will be presented to a conference on February 5.

The differential Galois theory of strongly normal extensions is ripe for study. It has been neglected, possibly because Kolchin used his own axiomatic definition of algebraic group. Instead, we use differential schemes, another area ripe for study.

We start with a sketch of Picard-Vessiot theory emphasizing its connection with tensor products. After defining strongly normal extensions, we show that an approach similar to that used for Picard-Vessiot theory also works for strongly normal extensions. However we must replace differential rings with differential schemes. This is not surprising as the Galois group is a group scheme that is not necessarily affine.

Strongly normal extensions are abundant; every connected group scheme is the Galois group of some strongly normal extension. And there is a "factory" to produce them - the logarithmic derivative. Yet explicit examples are difficult to find. There has been work on characterizing the type of equation needed, but much more is needed.

For lecture notes click here.

Friday, February 1, 2008

Richard Churchill, The Graduate Center and Hunter College
Differential Arithmetic

In this talk we indicate how elementary number-theoretic results depending on unique factorization can occasionally be established by "differentiating" integers. This presentation is intended to serve as an introduction to differential algebra, and is kept at an elemenary level: no background in that subject is assumed.

For lecture notes click here.

Friday, February 8, 2008

A. V. Mikhalev, Moscow State University
Derivations of conformal algebras
A. A. Mikhalev, Moscow State University
Derivations and primitive elements of free nonassociative algebras
Alexander Levin, The Catholic University of America
Compatibility of differential and difference field extensions

In this talk we consider the problem of compatibility in differential and difference algebra, that is, the problem of K-embedding of two differential (difference) field extensions L/K and M/K into some differential (respectively, difference) overfield of K. We are going to start with discussion of the compatibility of classical and differential field extensions. Then we will consider the situation in difference algebra where even two extensions of an ordinary difference field of zero characteristic can be incompatible. After introducing the concepts of the limit degree and core of a difference field extension and describing their properties, we will prove a criterion of compatibility for extensions of difference fields.

Friday, February 15, 2008

Li Guo, Rutgers University at Newark
Introduction to Rota-Baxter algebras

This is the first of a mini-series of four talks related to Rota-Baxter algebras. A Rota-Baxter operator is an abstraction and generalization of the integral operator in calculus by focusing on the integration by parts formula. A Rota-Baxter algebra is an algebra acted on by a Rota-Baxter operator. The study of Rota-Baxter algebras started with the work of Glenn Baxter in probability in 1960. Applications of Rota-Baxter algebras have been found in many areas recently in mathematics and physics. A differential Rota-Baxter algebra is a Rota-Baxter algebra with an added differentiation operator. In these introductory lectures, we give a brief history of Rota-Baxter algebras, a list of examples and basic properties. We will also discuss constructions of free objects in these categories.

Experts in these areas are encouraged to contact the organizers to arrange in-depth presentations on related topics.

For lecture notes click here.

Friday, February 22, 2008

Li Guo and William Keigher*, Rutgers University at Newark
Introduction to Differential Rota-Baxter algebras
*speaker

Friday, February 29, 2008

Li Guo and William Keigher*, Rutgers University at Newark
Introduction to Differential Rota-Baxter algebras (cont'd)

Li Guo, Rutgers University at Newark and William Sit*, City College of CUNY
Enumeration of Rota-Baxter and Differential Rota-Baxter Words
*speaker

For lecture notes click here.

Friday, March 7, 2008, 10:30 AM

Li Guo, Rutgers University at Newark and William Sit*, City College of CUNY
Enumeration of Rota-Baxter and Differential Rota-Baxter Words (cont'd)
*speaker

For lecture notes click here.

Friday, March 14, 2008, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Reversibles of ordinary linear differential equation

An ordinary linear differential equation defines a meromorphic connection over the Riemann spheres (and vice versa). Given a ramified covering of the sphere we can pull-back the connection to the covering surface. In this setting, the covering transforms can be lifted to the new connection as parallel automorphisms of the pull-back bundle. Such covering transforms are called reversibles. I will explain how, using the Galois group and the Fano group of a given connection, one can read if the connection is a pull-back of another.

Friday, March 28, 2008

Bernard Malgrange, Institut Fourier, Université de Grenoble

Friday, April 4, 2008

Raymond Hoobler, CCNY and The Graduate Center
Picard-Vessiot Extensions via Tannakian Categories

I will try to explain Deligne's approach to Picard-Vessiot extensions throught the Tannakian category formalism. This will provide an alternative view of affine differential Galois groups by studying their representations. An example due to Katz will be given as an illustration.

Friday, April 11, 2008

Bernard Malgrange, Institut Fourier, Université de Grenoble

Friday, April 18, 2008

Michael Zieve, Rutgers University
Bijections on rational points, with connections to difference fields.

Let V and W  be varieties over a finite field k. I will discuss morphisms f:V → W  over k for which the induced map on k-rational points  fk:V(k) → W(k) is bijective. When k is large relative to  f (and V), the Chebotarev density theorem (and some Galois theory) implies that any such  f satisfies strong structural constraints — essentially it behaves like a morphism of algebraic groups. When k is small,  fk can be bijective "at random". Intriguingly, in the intermediate range between the "random" and the Chebotarev situations, all known examples of bijective  fk's have a form resembling that of the  f's from the Chebotarev range, except that the expressions involve not just the coordinates of V but also their images under Frobenius. This suggests that these should be interpreted as morphisms of varieties over difference fields. I will describe some geometric properties of morphisms of algebraic groups that hold for all bijective  fk in the Chebotarev range; conjecturally, every "nonrandom" bijective  fk can be rewritten in terms of difference fields in such a way that these geometric properties are satisfied.

Friday, May 2, 2008

Michael Zieve, Rutgers University
Bijections on rational points, with connections to difference fields. Part II.


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